#120894

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,8,12,15)(4,7)(5,9)(6,10)(11,14,13,16)(17,18,23,25,22,19,21,26)(20,24,30,28,31,27,29,32);(2,7,10,9)(3,11,12,13)(8,15)(14,16)(17,19,22,18)(20,28)(21,25,23,26)(24,29)(27,30)(31,32);(1,3,4,11,6,12,5,13)(2,8,9,16,10,15,7,14)(17,20,21,29,22,31,23,30)(18,24,25,28,19,27,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28090);